{"id":2167,"date":"2023-04-28T16:40:02","date_gmt":"2023-04-28T16:40:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2167"},"modified":"2023-12-27T00:47:04","modified_gmt":"2023-12-27T00:47:04","slug":"modeling-exponential-growth-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/modeling-exponential-growth-fresh-take\/","title":{"raw":"Modeling Exponential Growth: Fresh Take","rendered":"Modeling Exponential Growth: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Perform exponential regression<\/li>\r\n\t<li>Convert between exponential and continuous growth<\/li>\r\n\t<li>Compare exponential and linear regressions for best fit<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> <br \/>\r\n<br \/>\r\n<p>Exponential growth occurs when a quantity grows in proportion to itself, such as in population growth or compound interest. This type of growth is characterized by rapid expansion over time.<\/p>\r\n<p><strong>Key Concepts:<\/strong><\/p>\r\n<ul>\r\n\t<li><strong>Exponential Regression<\/strong>: Fitting an exponential function to a set of data using software tools.<\/li>\r\n\t<li><strong>Continuous Growth<\/strong>: A form of exponential growth where the base of the exponential function is the irrational number [latex]e[\/latex] commonly found in natural growth processes.<\/li>\r\n<\/ul>\r\n<p><strong>Comparing Exponential and Continuous Growth Formulas:<\/strong><\/p>\r\n<ul>\r\n\t<li>Exponential Growth Formula: [latex]P_{n}=P_{0}(1+r)^{n} [\/latex]<\/li>\r\n\t<li>Continuous Growth Formula: [latex]y = ae^{rx}[\/latex]<\/li>\r\n\t<li>These formulas are used to model different types of growth scenarios, each with its unique characteristics.<\/li>\r\n<\/ul>\r\n\r\nUnderstanding how to convert between exponential and continuous growth models allows for flexibility in modeling various real-world scenarios.\r\n\r\n<p><strong>Steps for Conversion:<\/strong><\/p>\r\n<ol>\r\n\t<li><strong>Identify the Formulas<\/strong>: Recognize the exponential growth formula [latex]P_{n}=P_{0}(1+r)^{n} [\/latex] and the continuous growth formula [latex]y = ae^{rx}[\/latex].<\/li>\r\n\t<li><strong>Use Properties of Exponents<\/strong>: Apply the power rule of exponents to convert between the two forms.<\/li>\r\n\t<li><strong>Equating Bases<\/strong>: By equating the bases in the formulas, you can transform one model into the other.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p>Watch the following video for a demonstration of how to use a graphing calculator to obtain a regression formula and use it to make a prediction.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/QfBvEjBz_1s\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Perform+Exponential+Regression+on+a+Graphing+Calculator.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Perform Exponential Regression on a Graphing Calculator\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Perform exponential regression<\/li>\n<li>Convert between exponential and continuous growth<\/li>\n<li>Compare exponential and linear regressions for best fit<\/li>\n<\/ul>\n<\/section>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> <\/p>\n<p>Exponential growth occurs when a quantity grows in proportion to itself, such as in population growth or compound interest. This type of growth is characterized by rapid expansion over time.<\/p>\n<p><strong>Key Concepts:<\/strong><\/p>\n<ul>\n<li><strong>Exponential Regression<\/strong>: Fitting an exponential function to a set of data using software tools.<\/li>\n<li><strong>Continuous Growth<\/strong>: A form of exponential growth where the base of the exponential function is the irrational number [latex]e[\/latex] commonly found in natural growth processes.<\/li>\n<\/ul>\n<p><strong>Comparing Exponential and Continuous Growth Formulas:<\/strong><\/p>\n<ul>\n<li>Exponential Growth Formula: [latex]P_{n}=P_{0}(1+r)^{n}[\/latex]<\/li>\n<li>Continuous Growth Formula: [latex]y = ae^{rx}[\/latex]<\/li>\n<li>These formulas are used to model different types of growth scenarios, each with its unique characteristics.<\/li>\n<\/ul>\n<p>Understanding how to convert between exponential and continuous growth models allows for flexibility in modeling various real-world scenarios.<\/p>\n<p><strong>Steps for Conversion:<\/strong><\/p>\n<ol>\n<li><strong>Identify the Formulas<\/strong>: Recognize the exponential growth formula [latex]P_{n}=P_{0}(1+r)^{n}[\/latex] and the continuous growth formula [latex]y = ae^{rx}[\/latex].<\/li>\n<li><strong>Use Properties of Exponents<\/strong>: Apply the power rule of exponents to convert between the two forms.<\/li>\n<li><strong>Equating Bases<\/strong>: By equating the bases in the formulas, you can transform one model into the other.<\/li>\n<\/ol>\n<\/div>\n<p>Watch the following video for a demonstration of how to use a graphing calculator to obtain a regression formula and use it to make a prediction.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/QfBvEjBz_1s\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Perform+Exponential+Regression+on+a+Graphing+Calculator.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Perform Exponential Regression on a Graphing Calculator\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":15,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":8093,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2167"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2167\/revisions"}],"predecessor-version":[{"id":12856,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2167\/revisions\/12856"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/8093"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2167\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2167"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2167"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2167"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}